<p>The purpose of this article is to show a geometric version of Zabrodin–Wiegmann conjecture for an integer quantum Hall state. Given an effective reduced divisor on a compact connected Riemann surface, using the canonical holomorphic section of the associated canonical line bundle as well as certain initial data and local normalisation data, we construct a canonical non-zero element in the determinant line of the cohomology of the <i>p</i>-tensor power of the line bundle. When endowed with proper metric data, the square of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm of our canonical element is the partition function associated to an integer quantum Hall state. We establish an asymptotic expansion for the logarithm of the partition function when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\rightarrow +\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The constant term of this expansion includes the holomorphic analytic torsion and matches a geometric version of Zabrodin–Wiegmann’s prediction. Our proof relies on Bismut–Lebeau’s embedding formula for the Quillen metrics, Bismut–Vasserot and Finski’s asymptotic expansion for the analytic torsion associated to the higher tensor product of a positive Hermitian holomorphic line bundle.</p>